Recall that in our discussion of the mode equations for the hypercharge gauge field, we set the induced hyperelectric current to zero [see Eq. (REF )].
In order to properly account for the nonlinear backreaction of the fermion current on the efficiency of gauge-field production, we now need to revert this step and work with the explicit expression for the current \(\mathbf {J}\) .
In the case of our interest, where the hypercharge gauge field is strongly amplified, \(\left<\mathbf {E}^2\right> \sim \left<\mathbf {B}^2\right> \gg H^4\) , the typical length scale of fermion production, \(\left<\mathbf {E}^2\right>^{-1/4}\) , is much shorter than the correlation length of the hypercharge gauge field, \(H^{-1}\) .
Hence, one can approximate the hyperelectric and hypermagnetic fields as homogeneous and (anti) parallel, whose representative amplitudes are approximately given by \(E = \left<\mathbf {E}^2\right>^{1/2}\)  and \(B = \left<\mathbf {B}^2\right>^{1/2}\) , respectively.
Furthermore, we expect the hypercharge gauge field to reach a stationary configuration, where the tachyonic instability and the induced current balance each other, which implies that the physical amplitudes \(E/a^2\)  and \(B/a^2\)  become almost time-independent.
Under these approximations, the comoving fermion current was derived in Ref. [1]} as
\(\mathbf {J} = \frac{J}{E} \,\mathbf {E} \,,\qquad g_Y J = \frac{41g_Y^3}{72\pi ^2} \frac{EB}{a H} \coth \left(\pi \frac{B}{E}\right) \,.\) 
